As foundation Vision-Language Models (VLMs) unlock fine-tuning on smaller datasets while leveraging large-scale pre-training data, machine unlearning becomes critical in addressing privacy concerns and regulatory compliance. Task vector, representing the difference between parameters of models fine-tuned with and without specific data, is a popular retraining-free unlearning strategy. However, we observe that task vectors exhibit substantial sensitivity to various fine-tuning configurations, resulting in unstable unlearning effectiveness that correlates negatively with the prediction-level variance. While aggregating multiple functions (e.g., VLM with classifier) whose parameters are represented by different task vectors reduces function variance and improves unlearning, the computational cost of obtaining numerous task vectors and aggregating functions is computationally high. Thus, in order to capture the space of task vectors induced by diverse fine-tuning strategies, we propose modeling it within the convex hull of (Q 1)-simplex whose vertices represent Q task vectors. Although a function ensemble can be formed by sampling numerous task vectors from such a simplex, we derive a closed-form ensemble of an infinite number of functions whose parameters are uniformly sampled from the simplex, enabling efficient function-level task vector ensembling with enhanced unlearning performance. Extensive experiments and analyses across diverse datasets and scenarios demonstrate the efficacy of our method.

Vertex hunting (VH) is the task of estimating a simplex from noisy data points and has many applications in areas such as network and text analysis. We introduce a new variant, semi-supervised vertex hunting (SSVH), in which partial information is available in the form of barycentric coordinates for some data points, known only up to an unknown transformation. To address this problem, we develop a method that leverages properties of orthogonal projection matrices, drawing on novel insights from linear algebra. We establish theoretical error bounds for our method and demonstrate that it achieves a faster convergence rate than existing unsupervised VH algorithms. Finally, we apply SSVH to two practical settings-- semi-supervised network mixed membership estimation and semi-supervised topic modeling--resulting in efficient and scalable algorithms.

When a prediction algorithm serves a collection of users, disparities in prediction quality are likely to emerge. If users respond to accurate predictions by increasing engagement, inviting friends, or adopting trends, repeated learning creates a feedback loop that shapes both the model and the population of its users. In this work, we introduce evolutionary prediction games, a framework grounded in evolutionary game theory which models such feedback loops as natural-selection processes among groups of users. Our theoretical analysis reveals a gap between idealized and real-world learning settings: In idealized settings with unlimited data and computational power, repeated learning creates competition and promotes competitive exclusion across a broad class of behavioral dynamics. However, under realistic constraints such as finite data, limited compute, or risk of overfitting, we show that stable coexistence and mutualistic symbiosis between groups becomes possible. We analyze these possibilities in terms of their stability and feasibility, present mechanisms that can sustain their existence, and empirically demonstrate our findings.

Consider the task of locating an unknown target point using approximate distance queries: in each round, a reconstructor selects a reference point and receives a noisy version of its distance to the target. This problem arises naturally in various contexts--ranging from localization in GPS and sensor networks to privacy-aware data access--and spans a wide variety of metric spaces. It is relevant from the perspective of both the reconstructor (seeking accurate recovery) and the responder (aiming to limit information disclosure, e.g., for privacy or security reasons). We study this reconstruction game through a learning-theoretic lens, focusing on the rate and limits of the best possible reconstruction error. Our first result provides a tight geometric characterization of the optimal error in terms of the Chebyshev radius, a classical concept from geometry. This characterization applies to all compact metric spaces (in fact, even to all totally bounded spaces) and yields explicit formulas for natural metric spaces. Our second result addresses the asymptotic behavior of reconstruction, distinguishing between pseudo-finite spaces--where the optimal error is attained after finitely many queries--and spaces where the approximation curve exhibits a nontrivial decay. We characterize pseudo-finiteness for convex Euclidean spaces.

Transformers are effective at inferring the latent task from context via two inference modes: recognizing a task seen during training, and adapting to a novel one. Recent interpretability studies have identified from middle-layer representations task-specific directions, or task vectors, that steer model behavior. However, a lack of rigorous foundations hinders connecting internal representations to external model behavior: existing work fails to explain how task-vector geometry is shaped by the training distribution, and what geometry enables out-of-distribution (OOD) generalization. In this paper, we study these questions in a controlled synthetic setting by training small transformers from scratch on latent-task sequence distributions, which allows a principled mathematical characterization. We show that two inference modes can coexist within a single model. In-distribution behavior is governed by Bayesian task retrieval, implemented internally through convex combinations of learned task vectors. OOD behavior, by contrast, arises through extrapolative task learning, whose representations occupy a subspace nearly orthogonal to the task-vector subspace. Taken together, our results suggest that task-vector geometry, training distributions, and generalization behaviors are closely related.

We study stochastic multi-armed bandits in which the objective is a statistical functional of the long-run reward distribution, rather than expected reward alone. Under mild continuity assumptions, we show that the infinite-horizon problem reduces to optimizing over stationary mixed policies: each weight vector \(w\) on the simplex induces a mixture law \(P^w\), and performance is measured by the concave utility \(U(w)=\mathfrak U(P^w)\). For differentiable statistical utilities, we use influence-function calculus to derive stochastic gradient estimators from bandit feedback. This leads to an entropic mirror-ascent algorithm on a truncated simplex, implemented through multiplicative-weights updates and plug-in estimates of the influence function. We establish regret bounds that separate the mirror-ascent optimization error from the bias caused by estimating the influence function. The framework is developed for general concave distributional utilities and illustrated through variance and Wasserstein objectives, with numerical experiments comparing exact and plug-in influence-function implementations.

We develop new parameter-free and scale-free algorithms for solving convexconcave saddle-point problems. Our results are based on a new simple regret minimizer, the Conic Blackwell Algorithm+ (CBA+), which attains O(1/ T) average regret. Intuitively, our approach generalizes to other decision sets of interest ideas from the Counterfactual Regret minimization (CFR+) algorithm, which has very strong practical performance for solving sequential games on simplexes. We show how to implement CBA+ for the simplex, `p norm balls, and ellipsoidal confidence regions in the simplex, and we present numerical experiments for solving matrix games and distributionally robust optimization problems. Our empirical results show that CBA+ is a simple algorithm that outperforms state-ofthe-art methods on synthetic data and real data instances, without the need for any choice of step sizes or other algorithmic parameters.

We study Online Lazy Gradient Descent for optimisation on a strongly convex domain. The algorithm is known to achieve O( N) regret against adversarial opponents; here we show it is universal in the sense that it also achieves O(log N) expected regret against i.i.d opponents. This improves upon the more complex metaalgorithm of Huang et al [20] that only gets O( Nlog N) and O(log N) bounds. In addition we show that, unlike for the simplex, order bounds for pseudo-regret and expected regret are equivalent for strongly convex domains.

Strong mixed-integer programming formulations for trained neural networks.